Centrale-Supélec 2026 — Mathématiques 1 (PC)

Independent solutions, written by Pulsars. Neither official nor affiliated with Concours Centrale-Supélec.

1. Exercise 1 — Partie A · I — Équation réduite de l'ellipse (Q1 à Q5)

   F and F′ are the points with coordinates (c, 0) and (−c, 0) with 0 < c < a, b = √(a² − c²), and ℰ is the set of points M such that MF + MF′ = 2a.

   • Q1 — the triangle inequality, then FF′ ⩾ |MF − MF′|.
   • Q2 — deduce that MF − MF′ ± 2a does not vanish.
   • Q3 — an equivalence in the form of a product of four factors.
   • Q4 — the form ((MF′)² + (MF)² − 4a²)² = 4(MF′)²(MF)².
   • Q5 — the standard equation x²/a² + y²/b² = 1.
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2. Exercise 2 — Partie A · II — Étude d'une fonction (Q6 à Q8)

   f is defined on [0, a] by f(x) = b√(1 − x²/a²) — the upper right half of the ellipse.

   • Q6 — table of variations, and tangents to C_f at (0, b) and (a, 0).
   • Q7 — sketch of C_f.
   • Q8 — drawing of ℰ, with the vertices and the foci.
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3. Exercise 3 — Partie A · III — Paramètres de l'ellipse (Q9, Q10)

   We set e = c/a, the eccentricity.

   • Q9 — show e ∈ ]0, 1[, then express c and b in terms of a and e.
   • Q10 — shape of ℰ depending on whether e is close to 0 or to 1.
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4. Exercise 4 — Partie B — Mouvement des planètes massives (Q11 à Q15)

   Three planets, described by the vectors H_t and L_t, related by G_t = B·L_t and K_t = −B·H_t, where G_t and K_t are the vectors of derivatives.

   • Q11 — establish F_t = A·H_t with A = −B².
   • Q12 — diagonalisability of A, which has three distinct negative real eigenvalues.
   • Q13 — differential equation satisfied by each y_i, where Y_t = Q^{-1}H_t.
   • Q14 — general expression of the y_i.
   • Q15 — invertibility of B, then expression of the h_i and the l_i.
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5. Exercise 5 — Partie C · I — La méthode de Le Verrier (Q16 à Q19)

   S_1, S_2, S_3 denote the sums of the eigenvalues, of their squares and of their cubes, and P_A = α_0 + α_1X + α_2X² + α_3X³.

   • Q16 — Tr(AB) = Tr(BA), and the invariance of the trace under similarity.
   • Q17 — Tr(A), Tr(A²) and Tr(A³) in terms of the S_i.
   • Q18 — the characteristic polynomial and its coefficients in terms of the λ_i.
   • Q19 — α_1 and α_2 in terms of S_1 and S_2.
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6. Exercise 6 — Partie C · II — Étude d'un cas particulier (Q20 à Q23)

   A = ½·[[−4, 11, 13], [0, 1, 7], [0, 7, 1]] and R = X³ + X² − 14X − 24.

   • Q20 — triangularisation over C; do the relations of Q19 still hold?
   • Q21 — the three traces, and the characteristic polynomial.
   • Q22 — the eigenvalues.
   • Q23 — a change-of-basis matrix Q diagonalising A.
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7. Exercise 7 — Partie D · I — Perturbation de rang 1 : solutions et spectre (Q24 à Q29)

   D = diag(a, b, c) with a < b < c, U = (u, v, w) with non-zero coordinates, T = UU^⊤ and D(t) = D + tT. The equation (E_t) is

   u²/(λ−a) + v²/(λ−b) + w²/(λ−c) = 1/t.

   • Q24, Q25 — two identities satisfied by an eigenvalue λ of D(t).
   • Q26 — λ is a solution of (E_t).
   • Q27 — variations of F, and its two zeros ρ_1 < ρ_2.
   • Q28 — (E_t) has three distinct real solutions.
   • Q29 — the solutions of (E_t) and the spectrum of D(t) coincide.
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8. Exercise 8 — Partie D · II — Les limites (Q30 à Q34)

   g_1, …, g_4 denote the inverses of the restrictions of F to the four intervals ]−∞, a[, ]a, b[, ]b, c[ and ]c, +∞[.

   • Q30 — existence of the g_i, value of g_2(0), limits of g_2 at ±∞.
   • Q31 — the λ_i(t) in terms of the g_i.
   • Q32 — λ_1 extends by continuity to R.
   • Q33 — differentiability and value of λ_1′(0).
   • Q34 — limits of λ_1 at ±∞, and what this says about the perturbation.
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